Classification of cocompact discontinuous groups for G/H

Classify all pairs (G,H) of Lie groups for which the homogeneous space G/H admits cocompact discontinuous groups, namely discrete subgroups Gamma \subset G that act properly discontinuously and freely on G/H with compact quotient.

Background

The existence of cocompact discontinuous groups (compact CliffordKlein forms) for non-Riemannian homogeneous spaces is a long-standing problem intersecting geometry, topology, and representation theory.

Despite major progress (e.g., standard quotients and numerous obstructions), the full classification remains elusive even in seemingly simple cases such as rank-one symmetric spaces.